In 2008, Chen increased the packing density to .7786. A series of short-lived records were set and published prominently in Nature, Science, and Physical Review Letters, three of the most esteemed scientific journals to be published in. The New York Times reported on the results. All of these results have now been beaten.

As of July 27, 2010, the best-known packing density for the tetrahedron is 4000/4671 = .856347…. It was found by Chen, Engel, and Glotzer. Here’s Mathematica code for the 16-tetrahedra cell for this packing.

If tetrahedra aren’t regular, there are five known types that will fill space. These are listed in “Space-Filling Tetrahedra”. Here is the strangest of them, packed into a triangular prism:

I glossed over the mistakes made in finding these five polyhedra—three different mathematicians each found four of them. And there might be others, since this is an unsolved question. Even writing this blog entry seems foolhardy, since the best-known result is barely more than a month old. If I am wrong here, I get to join the ranks of 2,500 years worth of famous mathematicians, who were also wrong. But maybe the problem of tetrahedra packing is finally settled.

Back in 325 BC, Aristotle .. noted that regular tetrahedra could fill space.
Around 1470 AD, Regiomontanus showed that Aristotle was wrong.
– Wow, almost 2000 years to correct an obvious error. (obvious if you manually try to pack some regular tetrahedral)